momentum strategies under the carhart
model
arif memet - bachelor thesis
Student ID 10258264Supervisor Marijn Kool July 5th2014
Economics and Business
Specialization Economics and Finance Faculty of Economics and Business University of Amsterdam
A B S T R A C T
This paper examines the returns of Momentum Investment Strategies. Momen-tum Strategies buy stocks with high compounded returns. The paper uses strate-gies that vary the historical period and holding period over 1,2,3, 4 quarters. The results indicate that certain Momentum Strategies outperform the market and yield significant abnormal returns under the Carhart four-factor model. How-ever, the findings suggest that this is not a sign of market inefficiency.
1 introduction 1
2 literature 3
2.1 Momentum Strategies . . . 3
2.2 Contrarian Strategies . . . 5
2.3 Reconciling Momentum and Contrarian Views . . . 7
3 data and methodology 9
3.1 Data . . . 9
3.2 Strategy implementation . . . 11
3.3 Testing the strategies . . . 11
4 results 14
5 conclusion 16
Bibliography . . . 18
1
I N T R O D U C T I O N
Implementing a good investment policy is an essential factor to the success of any investor or finance professional. In the attempt to obtaining high returns, in-vestment funds, portfolio managers and individual investors implement various strategies. These strategies try to exploit market inefficiencies and mispricings or simply attempt to profit from the outcome of various stock-related events. Ultimately, every investment strategy is based on a set of fundamental beliefs about the behavior of investors and its effect on asset prices and markets. A momentum strategy is one type of such strategy that is used by a wide array of funds and finance professionals. Momentum is an observed tendency of a stock price to continue its recent movement trend. A stock with good historical performance should therefore continue to yield high returns, while the reverse should be true for stocks with poor historical performance. A momentum strat-egy tries to profit from this tendency by investing in stocks with high historical returns and selling stocks with low historical returns. Momentum strategies are used by a wide array of finance professionals. A particularly successful example of use of momentum is the example of the Value Line rankings (Porras & Gris-wold, 2009). Every week, Value Line Investment Survey publishes a survey of stock analysis, which contains a ranking of listed companies based on historical performance. These rankings have proved to be highly successful in selecting future winners and their success has yet to be explained, thusr resulting in the "Value Line enigma" (Porras& Griswold, 2009).
At the opposite side of the spectrum are contrarian strategies, which are based on the belief that stock markets overreact to information. Therefore an increasing stock price will continue to rise beyond its true value. Contrarian strategies consist in investing in bad performing stocks, in the belief that they are undervalued and selling good performing stocks.
This paper is focused on evaluating the performance of momentum strate-gies by testing for abnormal returns under the Carhart asset pricing model. The abnormal returns are a widely used measure for the analysis of a portfolio’s performance. As there are various past articles to support both the momentum and contrarian strategies, this paper also tries to reconcile these findings.
The structure of the paper is the following. The first section contains a list of previous findings on the subject, along with explanations of any theoretical concepts related to the topic. As contrarian and momentum strategies can be regarded as opposites, there will be different subsections discussing literature in favor of each strategy, along with an explanation of these different results. Furthermore, this section will contain a discussion of various momentum strate-gies and their performance measures. Section III will contain a discussion of the data and methodology used for this paper, along with any theoretical notions related to the methodology. The following section contains the results of the tests performed on each strategy. Finally, Section V will contain an interpreta-tion of the results secinterpreta-tion and their implicainterpreta-tions for Momentum Strategies and Market Efficiency.
2
L I T E R A T U R E
This section contains explanations of theoretical notions that are related to mo-mentum strategies, along with empirical findings that support the momo-mentum or contrarian views.
2.1
Momentum Strategies
One of the earliest papers on momentum investing was written by Levy (1967). In his paper, Levy uses weekly closing prices for NYSE stocks to calculate the ratio of the current week’s price to the average of the prices for the preceding 26 weeks. As this is a measure of a stock’s historical performance, he uses this ratio to construct a weekly ranking of relative strength containing all the stocks in his dataset. In order to test relative strength investments he constructs three other rankings: the market rank, which is the relative strength rank of the market portfolio, the volatility rank, which is based on the historical volatility of stocks, and the divergence rank, which is the weekly rank of the difference between the relative strength of the top performers and that of the bad performers. Levy’s (1994) results show that the divergence between top and bottom performers in-creases over time. Furthermore, the volatility ranks indicate that this divergence is even higher among the most volatile stocks. Levy’s results indicate that rel-ative strength is continued, with the gap between top and bottom performers increasing over time. Thus, his results favor the momentum view.
One of the most important papers on Momentum Strategies was written by Jegadeesh and Titman (1993). The authors discuss various findings related to the momentum and contrarian views. In their discussion they cast doubt on the findings of Levy (1967) by arguing that his paper might suffer from a selection bias in implementing the strategy. The authors also report that a different study suggests that on a different time horizon Levy’s strategy does not achieve abnormal returns (Jensen Bennington, 1970). After analyzing the
findings of previous papers, the authors provide their own methodology for evaluating Momentum Strategies. They use compounded returns for the past 1, 2, 3 and 4 quarters as measures of historical performance. Based on these measures, they divide the stocks into deciles and invest in the top decile. By also varying the holding period, they obtain 16 different strategy combinations. They use the Capital Asset Pricing Model to test for abnormal returns:
ri,t− rft= αi,t+ βi,t(rmkt,t− rf,t) + εi,t
where αi is the abnormal return, or the part of the return that is not ex-plained by the variation in the market return, βiis the market sensitivity and εi is an error term. They proceed to test the significance of αi for the 6 month/ 6 month strategy, which was the strategy with the highest excess returns. Their results show that αiis statistically significant. However, this does not necessarily imply that this strategy yields a riskless profit. The authors (Jegadeesh Titman, 1993) suggest that in the case that the error terms are not correlated, the abnor-mal returns are the result of taking a risk. However, the reverse would indicate to market inefficiency. The performance of momentum stocks is summarized in the following equations:
E(ri,t−¯rt|ri,t−1−¯rt−1 > 0) > 0 E(ri,t−¯rt|ri,t−1−¯rt−1 < 0) < 0
Therefore, stocks that had a higher return than average in the previous period are expected to have a higher-than-average return in the current period and the reverse should be true for lower-than-average performances. From the above, the following equation can be derived:
E[(ri,t−¯rt)(ri,t−1−¯rt−1)] > 0
Based on the CAPM equation, this results in the following:
E[(ri,t−¯rt)(ri,t−1−¯rt−1)] = σ2α+ σ2βCov(rmkt,t, rmkt,t−1) +Cov¯ i(ei,t, ei,t−1) By calculating a positive serial covariance for the error terms ¯Covi(ei,t, ei,t−1) and a negative serial covariance for the market returns, Cov(rmkt,t, rmkt,t−1)the authors conclude that the returns of momentum stocks are caused by a market underreaction to firm-specific information, thus indicating to a market ineffi-ciency.
2.2 contrarian strategies 5 Building up on the work of Jegadeesh and Titman (1993), Grundy and Martin (2001), replicate their strategy implementation while testing against the Fama-French 3-factor model:
ri− rf = αi+ βirmkt + siSMB + hiHML + εi
In this model, the additional factor SMB stands for Small-Minus-Big and measures the excess returns of small market-capitalization stocks over high market-capitalization stocks. The factor HML stands for High-Minus-Low and represents the excess returns of small price-to-book ratio stocks over large ones. The small P/B stocks are considered to be value stocks while the large ones are growth stocks. The results of the test show that the Fama-French model also yields significant abnormal returns. Following this analysis, the authors (Grundy Martin 2001) argue that part of the high returns of momentum stocks are likely to be caused by momentum in the Fama-French factors. Therefore, they propose a different stock-selection procedure, which is based on the of each stock and should thus select stocks based on idiosyncratic momentum. Af-ter implementing this strategy and testing against the Fama-French model, the paper reports significant abnormal returns, while maintaining a lower return variance.
Several other papers have been written on Momentum Strategies and yield mixed results. Porras and Griswold (2009) attempt to explain the performance of the top firms in the Value Line rankings, a weekly ranking realized by Value Line Investment Survey, which has proved to be highly successful at finding firms with future high returns. After using various multi-factor models to ex-plain the returns of top performers, the authors consistently find significant abnormal returns, thus favoring the momentum view. Moskowitz and Grinblatt (1999) use a model containing a series of industry factors to explain momentum-strategy returns. Their results show that when controlling for industry factors, there are no significant abnormal returns.
2.2
Contrarian Strategies
An extensive literature has been written on Contrarian Strategies. As an implementation of a contrarian strategy is the opposite of a momentum strategy, papers on the contrarian view are also relevant to the momentum view.
One of the most important papers on Contrarian Investing was written by Lakonishok et al. (1994). The authors use the term “glamour stocks” to describe stocks with high recent returns and “value stocks” to describe stocks with low recent returns. They argue that the market over-reacts to informa-tion and thus the “glamour stocks” are overpriced while the “value stocks” are underpriced. One of the criticisms of Contrarian Strategies comes from Fama and French (1992), who argue that “value stocks” achieve high returns due to the fact that they are riskier. In consequence, the paper of Lakonishok et al. (1994) aims to explain the returns of Contrarian Strategies, as well as investi-gate whether these returns are indeed riskier. A series of ranks are constructed using book-to-market ratio, cash flow-to-price ratio and earnings-to-price ratio. These rankings are supposed to separate “value stocks” from “glamour stocks”. The paper suggests that over a period of 5 years, “value stocks” significantly outperform “glamour stocks”. Furthermore, the authors perform a regression analysis using a multi-factor model. Their results show that the model is suc-cessful in predicting the contrarian returns. Finally, the authors (Lakonishok et al., 1994) are unable to reject the claim that contrarian returns are high because of their fundamental risk. However, they argue that given the significant abnor-mal returns they found, and the difference in performance between value and glamour stocks, value stocks are underpriced for the risk they offer.
De Bondt and Thaler (1985) review a series of papers in Psychology and Economics which come to the consensus that stock markets have the tendency to overreact to information. Based on this information, they develop a set of two assumptions. First, extreme movements in stock prices should be reversed in the following period. Secondly, a more extreme price movement should result in a greater magnitude of the reversal. Their results favor the contrarian view and indicate to a stock market overreaction. Moreover, the overreaction is found to be much larger for losers than for winners.
Chen and Zhang (1998) also analyze the returns of value stocks and try to establish common risk characteristics of these stocks. They analyze the differ-ent behavior of stock returns in six differdiffer-ent countries. By using a multi-factor model they manage to explain the variation of stock returns through the vari-ation of the factors. Their results also suggest that the high returns of value stocks are a good compensation for their risk. Their cross-country analysis sug-gests that value stocks have higher returns in the US than in the rest of the countries in their sample. The authors suggest that this is due to the fact that these returns come from firms that have a high leverage and uncertain future earnings, and are therefore more volatile.
2.3 reconciling momentum and contrarian views 7
2.3
Reconciling Momentum and Contrarian Views
After reviewing various papers related to both Momentum Strategies and Con-trarian Strategies, it is apparent that previous literature provides mixed evi-dence with regard to the topic of this paper. In order to reconcile some of the differences in the conclusions of these papers, some of the differences in their methodologies are discussed below.
First of all, the debate of Momentum vs Contrarian views has taken place over a long period of time. Most of the articles considered in this paper are writ-ten in different periods, and therefore have also different time horizons in their data set. Moreover, differences may arise due to the stock selection pool chosen for the implementation of the strategy. As these pools vary, this could also lead to different results. Another important factor related to data is a potential bias in stock selection. As most of the literature reviewed in this paper considers stocks from the New York Stock Exchange or American Stock Exchange, the cost pools used for the strategies are very large. In addition they often contain very volatile stocks that are relatively small, or newly listed. As many of these stocks will be part of the top decile in historical returns, it is also reasonable to expect that their performance will reverse, thus causing a downward bias in the performance of momentum stocks.
Other differences in the results of past papers are likely to come from different strategy implementations. Even though most papers are set to test the same sets of fundamental beliefs about stock market reactions, the choice of strategy is relevant to the results. In this sense, evidence that supports the contrarian view does not necessarily contradict the evidence for momentum strategies. Some of the most influential contrarian papers, such as the one by Lakonishok et al. (1994) and De Bondt and Thaler (1985) use a historical time period of 3-5 years to rank the stocks based on performance. In addition these papers also use 3-5 years holding periods for their strategies. In contrast, most momentum papers, such as those of Jegadeesh and Titman (1993) and Grundy and Martin (2001) use 3-12 month historical and holding periods. This differ-ence suggests that it is likely that stock market underreacts in the sort run and overreacts in the long run, thus favoring the short run momentum view and long run contrarian view.
Other differences may arise due to differences in performance measures that are used for both stock selection and return analysis. Lakonishok et al.
(1994) use a series of financial ratios to detect value stocks. Similar to De Bondt and Thaler (1985), they compare the returns of the top decile with that of the lower decile. In contrast, the papers of Jegadeesh and Titman (1993) and Grundy and Martin (1993) use returns as a measure of stock performance for both stock selection and evaluation. Finally, there are also differences in the statistical tests performed, with some papers testing for a difference in mean between the top and lower decile and other papers focusing on testing the top decile against a factor model, such as the CAPM or the Fama-French model.
3
D A T A A N D M E T H O D O L O G Y
This section is concerned with discussing the data and methodology that will be used for this paper along with any theoretical and statistical matters that are relevant to strategy or test implementation. Reviewing the differences in other papers’ results and methodologies has led to the selection of the data and methodology for this paper, as it attempts to avoid certain biases and other problems.
3.1
Data
This paper uses data for all companies in the composition of the S&P500 index. As it is apparent from past papers that using a large stock pool with many small and volatile stocks can lead to a downward bias in the future performance of the top decile, this paper aims to remove this bias. By using the S&P500 as a stock selection pool, the small and volatile stocks will be avoided and there should be no bias for the top decile. An alternative method could be to invest in the top decile based on market capitalization. Since the volatile stocks are generally small, they will not have a significant effect on the performance of the momentum portfolio. However, as the methodology for most previous papers suggests, the stocks are to be divided into deciles, based on their performance. This division does not take into account market capitalization and can lead to a top decile that is composed of a large number of small volatile stocks, without regard to their size. Therefore the momentum portfolio performance might be driven by only a few stocks and that in turn could lead to other biases.
Based on these considerations, daily prices for S&P500 were collected us-ing Datastream. These prices are adjusted to include any dividends and share repurchases. The dividends and share repurchases tend to drive stock prices
down and create a downward bias, as they do not reflect the fact that their pro-ceeds can be reinvested. The following step is to compute returns based on the adjusted prices:
rt+1 = (Pt+1− Pt)/Pt
There are several issues to consider with regard to selecting S&P500 stocks. First of all, the composition of the index changes over time. This paper consid-ers the index constituents as of June 2014 and records return for the period 01/01/2004 to 01/01/2014. This results in two problems regarding the stock pool. First of all, there are a lot of companies that were constituents of the index during the period considered and their returns have not been recorded. This should not be an issue for the analysis at hand, as stock data for this paper is instrumental in obtaining the top returns decile from a large pool of stocks. As long as there is no bias in the stock or stock pool selection, the composition of the stock pool is not relevant to the analysis. The second problem is that some of the companies included in the stock pool were not listed for the entire period considered. In consequence, for a large part, there will not be enough data for these stocks to be considered for inclusion in the top decile. In order to deal with this issue, the returns for the periods where stock data is missing have been replaced with -1. This is the equivalent of the company going bankrupt or being liquidated. Since the stock pool contains the composition of the index at a time later than the period considered, none of the stocks have been liqui-dated. Therefore, the -1 return stocks will be assigned to the lower decile for the periods where data is missing, thus not influencing the momentum portfo-lio. Another possible issue is the fact that the stock pool is not as large as in most other papers. Although the NYSE is a considerably larger stock pool, the S&P500 has been selected in order to deal with the selection bias in other papers. The top decile will therefore be composed of 50 large companies. These issues do not affect the statistical significance of the tests performed, due to the fact that the sample size remains large – daily returns for a 10 year period and there is no missing data for the momentum portfolio – the top decile.
In addition to stock prices, data for the Fama-French factors and a momen-tum factor has been collected from the Kenneth French Data Library. This data is to be used for testing the momentum portfolio against a factor model. In addition, weekly rates for US Treasury Bills have been considered. These have been recorded as daily returns that have a weekly compounded return equal to the weekly T-bill rate.
3.2 strategy implementation 11
3.2
Strategy implementation
The first step in reaching a conclusion with regard to momentum investing is the choice of strategy implementation. This paper follows the methodologies of Jegadeesh and Titman (1993) and Grundy and Martin (2001) for the stock selection. Different periods of 1, 2, 3 or 4 quarters have been used to calculate compounded returns. These returns will be used as a reflection of historical performance. Therefore, the stocks will be divided into deciles based on com-pounded returns. The best performing stocks are considered to be momentum stocks. On average, the momentum view holds that they will continue to yield high returns. Therefore, the top decile will be considered as the momentum portfolio. This strategy is based on investing in the top decile and varying the holding period of current stocks for 1, 2, 3 or 4 quarters. Investing in the top decile is made using equal weights for each company, thus 1/50. This choice was preferred to assigning weights based on market capitalization in order to obtain various significant sources of momentum portfolio returns. Assigning weights based on market capitalization leads to high weights for large stocks and can therefore create biased results. The implementation of the strategy is re-alized using a C++ algorithm. The code for the algorithm along with comments on implementation steps are included in the Appendix section. The general approach is to compute compounded returns for each stock at the beginning of each holding period. For example, for a 3 months/3 months strategy, at the beginning of the fourth month in the sample, the compounded returns for the first, or previous 3 months will be calculated for all stocks. The stocks will then be ranked based on these returns, and the first 50 stocks represent the top decile. These stocks are assigned equal weights of 1/50. For the duration of the holding period – months 4-6 the average return of these stocks will be recorded as the daily momentum portfolio return. At the end of each holding period, the same procedure is implemented, until the end of the sample is reached.
3.3
Testing the strategies
As a result of the strategy implementation, the daily returns for every combi-nation of momentum portfolio are obtained. There are a few performance mea-sures that are of interest in evaluating the return performance. First, the returns
for the momentum portfolio will be compared with the returns of the market portfolio. This should reflect whether a momentum strategy will outperform a simple buy-and-hold strategy.
The following step is to regress the portfolio returns against the Carhart model. This is an improvement to the methodology of most of the previous papers, which used the Fama-French model.
The 3-factor model was proposed by Fama and French (1993) as an exten-sion of the Capital Asset pricing model. Following their work, Carhart (1997) realizes a further expansion of the model by adding a momentum factor. In the equation above the following terms are defined:
Ri= ri− rf is the excess return of an individual stock.
Rmkt = rmkt− rf is the excess return of the market portfolio and βi is the corre-sponding coefficient.
SML stands for Small-Minus-Big and represents excess returns of Small Market Capitalization stocks over Large Stocks.
HML stands for High-Minus-Low and represents excess returns of high book-to-market ratio stocks (value stocks) over low B/M stocks (growth stocks). MOM is the momentum factor and represents excess returns of high return stocks over low return stocks. This factor has been included for the analysis as it represents the same assumptions as the Momentum Strategies.
εi is an error term.
Finally, αi is the performance measure chosen. It is the abnormal return, or the part of the stock’s return that is not explained by the variation in the factors. As the MOM is included in the model, evaluating stocks based on α should reveal whether the momentum of the portfolio is fully explained by the momentum in the market, along with the accompanying factors. In case of a significant α the results would indicate to idiosyncratic momentum in the stocks (i.e. momen-tum that is specific to the selected stocks and cannot be explained by market movements). Therefore, if α is significant, it can be concluded that the strategy is likely to be profitable when hedging against the four Carhart factors.
However, obtaining a significant α would demand a more comprehensive interpretation. Even though a significant α indicates to a profitable strategy with
3.3 testing the strategies 13 returns that are not fully explained by the model, the source of these returns remains uncertain. Following the approach of Jegadeesh and Titman (1993), the possible sources in the returns are summarized in the following equation:
E[(ri,t−¯rt)(ri,t−1−¯rt−1)] = σ2α+ σ2βCov(rmkt,t, rmkt,t−1) + σ2sCov(SMLt, SMLt−1) + σ2hCov(HMLt, HMLt−1)+
σ2mCov(MOMt, MOMt−1) +Cov(ε¯ i,t, εi,t−1)
Therefore, if any of the serial covariances for the Carhart factors are found to be positive, it can be inferred that part of the returns are due to an exposure to the respective factors. However, the measure of interest here isCov¯ i(εi,t, εi,t−1) which is the serial covariance of the error terms. If there is a positive serial co-variance or autocorrelation in the error terms, it is likely that part of the returns are due to an underreaction by the market. This would therefore indicate to a market inefficiency. Thus, the weak form market efficiency hypothesis, which states that all information is reflected in the stock price (Bodie, Kane Marcus, 2011, p. 347), will be rejected.
4
R E S U L T S
This section contains a presentation of the findings resulted from applying the methodology described in the previous section. Four strategies out of the 16 possible combination have been selected for analysis. Therefore, the strategies with the historical period equal to the holding period have been selected (1 quarter historical period - 1 quarter holding period, etc.). They are henceforth referred to as One-One, Two-Two, Three-Three and Four-Four.
As described in the methodology, first, the algorithm for selecting the stocks based by historical return has been applied. The returns of the result-ing portfolios, along with the return for the market have been summarized in the following table:
Table 1:Returns
Strategy Average daily return Cumulative return
One-one 0.073% 325%
Two-two 0.081% 377%
Three-Three 0.068% 256%
Four-Four 0.069% 264%
Market 0.039% 112%
As it is apparent from the above table, the four portfolios obtain signifi-cantly higher returns than the market, with a 0.081% average return and 377% cumulative return for the top performing portfolio and a 0.039% average return and a 112% cumulative return for the market. These results, therefore, suggest that the Momentum Strategies outperform a simple buy-and-hold strategy by a significant margin.
The next step in the analysis is to test these returns under the Fama-French model. Most of the previous literature suggests that the Momentum Strategies should yield significant abnormal returns under this factor model. The outcome of the regressions is summarized in the table below:
results 15
Table 2:Fama-French regressions
Strategy Alpha t P>|t| One-one 0.000276 1.86 0.064 Two-two 0.00072 2.03 0.042 Three-Three 0.000609 1.66 0.097 Four-Four 0.00063 1.81 0.07
The regression outcomes suggest that all the strategies yield significant alphas at a 10% confidence level. However, only the Two-Two strategy yields abnormal returns for a 5% confidence level, with a t-value of 2.03. These find-ings are consistent with the previous table, as the Two-Two strategy was the top performer of the four.
As the Two-Two portfolio yields unexplained returns under the Fama-French model. The next step of the analysis is to test the strategies under the Carhart Model, which also contains the explanatory variable Momentum. The following table summarizes the regression output and also includes autocorre-lation results where necessary:
Table 3:Carhart regressions
Strategy Alpha t P>|t| Autocorrelation One-one 0.000226 1.64 0.102
Two-two 0.000738 2.09 0.037 -0.045 Three-Three 0.000601 1.64 0.101
Four-Four 0.000635 1.82 0.068 -0.0431
As can be expected, the Carhart model has a higher explanatory power due to the additional factor. Therefore the One-One and Three-Three portfo-lios no longer yield significant abnormal return at a 10% confidence level. The Four-Four strategy still has a significant α at 10% with a t-value of 1.82, while the Two-Two strategy has a significant α at 5% with a t-value of 2.09. As these two strategies still yield abnormal returns under the Carhart model, the auto-correlation of the regression residuals has been calculated for both portfolios. As mentioned in the Methodology section, a positive autocorrelation term leads to a rejection of the Weak-Form Market Efficiency Hypothesis. As both the au-tocorrelation terms are negative (-0.045 and -0.0431), the findings suggest that the returns of these strategies are indicative of a higher exposure to the Carhart factors.
5
C O N C L U S I O N
This paper aims to test the profitability of momentum investing. Momentum strategies are considered to be strategies that select the top performing stocks in terms of recent historical performance. As defined in this paper, the indicators of historical performance can be either the cumulative return or the α under the Carhart Model. The latter is considered in order to isolate the idiosyncratic momentum of each stock and use it build the portfolio.
In the first stage, a combination of 16 different strategies was implemented by varying the historical period and the holding period. Four strategies were selected from the 16 and have been analysed. The results show that the Momen-tum portfolios significantly outperform the market, with a 377% compounded return for the Two-Two portfolio. Furthermore, regressing the portfolio returns against the Fama-French and the Carhart models has shown that the Two-Two portfolio continues to hold significant abnormal returns at a 5% confidence level. A preliminary conclusion to the research purpose of this paper would be that a certain form of Momentum Strategy can indeed offer abnormal returns under the four-factor Carhart model. However, as the autocorrelation term for the strategy was found to be negative, this result is not indicative of a market imperfection. The return is thus caused by an exposure to the Carhart Factors.
There are several limitations and possible expansions to this research. First of all, the performance of Momentum Portfolios is likely to be affected by the business cycles. Particularly during recessions investors are expected to over-react to information(Garcia, 2013) thus enhancing the Momentum effect in the market. Data from the National Bureau of Economic Research suggests that during the time period considered by this paper (2004-2014) there was only one recession between 2007 and 2009. In this regard, the results of this paper might contain a bias, as momentum tends to increase during recession. Further research could thus be performed taking this aspect into consideration. Further-more, as the methodology of Grundy and Martin(2003) suggests, it is possible to implement a series of Momentum Strategies that select stocks based on their
conclusion 17 individual (α over the historical period. This choice of performance measure should thus select stocks based on idiosyncratic momentum. This is due to the fact that the performance measure - the α is not correlated with any of the carhart factors. Other possible improvements are using a larger stock pool while eliminating the stock-selection bias and using a longer time-period.
Bibliography
Bodie, Z., Kane, A., Marcus, A.J. (2011) Investments, 9th edition. New York: Mcgraw-Hill/Irwin
Bondt, W. F., Thaler, R. (1985). Does the stock market overreact?. The Journal of finance, 40(3), 793- 805.
Chen, N. F., Zhang, F. (1998). Risk and Return of Value Stocks*. The Journal of Business, 71(4), 501- 535.
Fama, E. F., French, K. R. (1992). The crosssection of expected stock returns. The Journal of Finance, 47(2), 427-465.
French, K., (2014) Fama/French Factors [Daily], Momentum Factor (Mom) [Daily]. Retrieved from http://mba.tuck.dartmouth.edu/pages/faculty/ ken.french/data_library.html#Research
Garcia, D. (2013). Sentiment during recessions. The Journal of Finance, 68(3), 1267-1300.
Grundy, B. D., Martin, J. S. (2001). Understanding the nature of the risks and the source of the rewards to momentum investing. Review of Financial Stud-ies,14(1), 29-78.
Jegadeesh, N., Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. The Journal of Finance, 48(1), 65-91. Jensen, M. C., Benington, G. A. (1970). Random walks and technical theories:
Some additional evidence. The Journal of Finance, 25(2), 469-482.
Lakonishok, J., Shleifer, A., Vishny, R. W. (1994). Contrarian investment, extrap-olation, and risk. The journal of finance, 49(5), 1541-1578.
Levy, R. A. (1967). Relative strength as a criterion for investment selection. The Journal of Finance, 22(4), 595-610
Moskowitz, T. J., Grinblatt, M. (1999). Do industries explain momentum?.The Journal of Finance, 54(4), 1249-1290.
NBER, (2014) US Business Cycle Expansions and Contractions. Retrieved from http://www.nber.org/cycles.html
Porras, D., Griswold, M. (2000). The Value Line Enigma Revisited. Quarterly Journal of Business and Economics, 39-50.
Stock, J. H., Watson, M. W. (2012). Introduction to econometrics, Third Edition . Harlow: Pearson Education Limited.
A P P E N D I X
Appendix A – C++ algorithm for 3 month/3month compounded
return strategy
\#include<fstream.h> \#include<iostream.h> \#include<math.h>
//y[i] - historical compounded return //x[i] - working variable - daily returns //v[i][j] - working variable - monthly returns //order - initial order of the stocks
//n - current no of months - start of historical to the end of holding period
//b, bb - control variables //s - daily portfolio return
float start,y[502],aux,v[502][20],x[502],weight[502],order[502],s; int b,bb,i,n,day,year,j;
void ranks() {
//Sort by compounded 3month return bb=1; while(bb==1) { bb=0; for(i=2;i<=498;i++) if(y[i]<y[i+1]) { bb=1; 19
aux=y[i]; y[i]=y[i+1]; y[i+1]=aux; aux=order[i]; order[i]=order[i+1]; order[i+1]=aux; } }
//Assign equal weights(0.02) to the top decile
for(i=2;i<=499;i++)if(i<=51)weight[i]=0.02; else weight[i]=0;
//Sort back to the initial stock order bb=1; while(bb==1) { bb=0; for(i=2;i<=498;i++) if(order[i]>order[i+1]) { bb=1; aux=y[i]; y[i]=y[i+1]; y[i+1]=aux; aux=order[i]; order[i]=order[i+1]; order[i+1]=aux; aux=weight[i]; weight[i]=weight[i+1]; weight[i+1]=aux; } } } void main() {
//input file - daily returns ifstream in("returns.txt");
21 //output file - daily returns*weight ; daily returns for
portfolio ofstream out("out1q1q.txt"); n=1; b=0; start=1; in>>x[1]>>day>>year; v[1][n]=x[1]; for(i=2;i<=499;i++) { v[i][n]=0; in>>x[i]; }
out<<"month day year ";
for(i=1;i<=498;i++)out<<i<<" "; out<<"sum"<<endl; while(!in.eof()) { while(x[1]==v[1][n]) { for(i=2;i<=499;i++) v[i][n]=(1+x[i])*(1+v[i][n])-1; in>>x[1]>>day>>year; x[1]=x[1]+12*(year-2004); for(i=2;i<=499;i++) in>>x[i]; if(b==1){
out<<x[1]<<" "<<day<<" "<<year<<" "; s=0;
for(i=2;i<=499;i++) { s+=x[i]*weight[i]; out<<x[i]*weight[i]<<" "; } out<<s<<endl; } }
//Start implementing the strategy from month 4
if(x[1]==4)b=1;
//At the beginning of a new holding period if(b==1&&start+3==x[1])
{
//Reset historical return compounding for(i=1;i<=499;i++) {y[i]=0;order[i]=i;} for(i=1;i<=3;i++) for(j=2;j<=499;j++) y[j]=(1+y[j])*(1+v[j][i])-1; ranks(); start=start+3;
//Delete unnecessary data from month vector v n=n-3; for(i=1;i<=n;i++) for(j=1;j<=499;j++) v[j][i]=v[j][i+3]; for(i=n+1;i<=n+3;i++) for(j=1;j<=499;j++) v[j][i]=0; } //New month
23 n++; for(i=1;i<=499;i++) v[i][n]=x[i]; } in.close(); out.close(); }
Regression and Autocorrelation outputs
Table 4:One-One Fama-French
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF 1.09607 0.012899 84.97 0 1.070775 1.121364 HML 0.072214 0.028617 2.52 0.012 0.016098 0.12833 SMB 0.270115 0.027112 9.96 0 0.21695 0.32328 _cons 0.000276 0.000149 1.86 0.064 -1.6E-05 0.000568
Table 5:One-One Carhart
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF 1.15661 0.01236 93.58 0 1.132373 1.180846 HML 0.308583 0.029122 10.6 0 0.251476 0.365689 SMB 0.229675 0.025255 9.09 0 0.180152 0.279199 MOM 0.324529 0.016369 19.83 0 0.292431 0.356628 _cons 0.000226 0.000138 1.64 0.102 -4.5E-05 0.000497
Table 6:Two-Two Fama-French
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF -0.01306 0.0306862 -0.43 0.671 -0.0732288 0.047118 SMB 0.046344 0.064497 0.72 0.472 -0.0801306 0.172818 HML 0.134336 0.0680775 1.97 0.049 0.0008401 0.267831 _cons 0.00072 0.0003543 2.03 0.042 0.0000251 0.001414
Table 7:Two-Two Carhart
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF -0.03487 0.031624 -1.1 0.27 -0.0968839 0.027141 HML 0.049159 0.0745135 0.66 0.509 -0.0969573 0.195275 SMB 0.060916 0.0646188 0.94 0.346 -0.0657969 0.18763 MOM -0.11695 0.0418823 -2.79 0.005 -0.1990744 -0.03482 _cons 0.000738 0.0003538 2.09 0.037 0.000044 0.001432
Table 8:Two-Two Autocorrelation
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] 1 -0.0450 -0.0450 4.7575 0.0292
25
Table 9:Three-Three Fama-French
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF 0.024374 0.031736 0.77 0.443 -0.03786 0.086606 HML 0.031458 0.070407 0.45 0.655 -0.1066 0.169521 SMB 0.022394 0.066703 0.34 0.737 -0.10841 0.153195 _cons 0.000609 0.000366 1.66 0.097 -0.00011 0.001327
Table 10:Three-Three Carhart
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF 0.034533 0.032747 1.05 0.292 -0.02968 0.098749 HML 0.071124 0.07716 0.92 0.357 -0.08018 0.22243 SMB 0.015608 0.066914 0.23 0.816 -0.11561 0.146822 MOM 0.05446 0.04337 1.26 0.209 -0.03059 0.139506 _cons 0.000601 0.000366 1.64 0.101 -0.00012 0.001319
Table 11:Four-Four Fama-French
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF 0.008915 0.029907 0.3 0.766 -0.04973 0.067561 SMB -0.04256 0.062985 -0.68 0.499 -0.16607 0.080948 HML -0.08723 0.066288 -1.32 0.188 -0.21721 0.042761 _cons 0.00063 0.000348 1.81 0.07 -5.2E-05 0.001313
Table 12:Four-Four Carhart
Excess Coef. Std. Err. t P>t [95% Conf. Interval] MktRF 0.003158 0.030917 0.1 0.919 -0.05747 0.063785 HML -0.10902 0.072617 -1.5 0.133 -0.25142 0.033379 SMB -0.03885 0.063193 -0.61 0.539 -0.16277 0.085073 MOM -0.03009 0.040917 -0.74 0.462 -0.11032 0.050147 _cons 0.000635 0.000348 1.82 0.068 -4.8E-05 0.001318
Table 13:Four-Four Autocorrelation
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] 1 -0.0431 -0.0431 4.4548 0.0348
